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from sympy.core.function import (Derivative, Function, diff)
from sympy.core.mul import Mul
from sympy.core.numbers import (Integer, pi)
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.trigonometric import sin
from sympy.physics.quantum.qexpr import QExpr
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.hilbert import HilbertSpace
from sympy.physics.quantum.operator import (Operator, UnitaryOperator,
HermitianOperator, OuterProduct,
DifferentialOperator,
IdentityOperator)
from sympy.physics.quantum.state import Ket, Bra, Wavefunction
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.spin import JzKet, JzBra
from sympy.physics.quantum.trace import Tr
from sympy.matrices import eye
from sympy.testing.pytest import warns_deprecated_sympy
class CustomKet(Ket):
@classmethod
def default_args(self):
return ("t",)
class CustomOp(HermitianOperator):
@classmethod
def default_args(self):
return ("T",)
t_ket = CustomKet()
t_op = CustomOp()
def test_operator():
A = Operator('A')
B = Operator('B')
C = Operator('C')
assert isinstance(A, Operator)
assert isinstance(A, QExpr)
assert A.label == (Symbol('A'),)
assert A.is_commutative is False
assert A.hilbert_space == HilbertSpace()
assert A*B != B*A
assert (A*(B + C)).expand() == A*B + A*C
assert ((A + B)**2).expand() == A**2 + A*B + B*A + B**2
assert t_op.label[0] == Symbol(t_op.default_args()[0])
assert Operator() == Operator("O")
with warns_deprecated_sympy():
assert A*IdentityOperator() == A
def test_operator_inv():
A = Operator('A')
assert A*A.inv() == 1
assert A.inv()*A == 1
def test_hermitian():
H = HermitianOperator('H')
assert isinstance(H, HermitianOperator)
assert isinstance(H, Operator)
assert Dagger(H) == H
assert H.inv() != H
assert H.is_commutative is False
assert Dagger(H).is_commutative is False
def test_unitary():
U = UnitaryOperator('U')
assert isinstance(U, UnitaryOperator)
assert isinstance(U, Operator)
assert U.inv() == Dagger(U)
assert U*Dagger(U) == 1
assert Dagger(U)*U == 1
assert U.is_commutative is False
assert Dagger(U).is_commutative is False
def test_identity():
with warns_deprecated_sympy():
I = IdentityOperator()
O = Operator('O')
x = Symbol("x")
three = sympify(3)
assert isinstance(I, IdentityOperator)
assert isinstance(I, Operator)
assert I * O == O
assert O * I == O
assert I * Dagger(O) == Dagger(O)
assert Dagger(O) * I == Dagger(O)
assert isinstance(I * I, IdentityOperator)
assert three * I == three
assert I * x == x
assert I.inv() == I
assert Dagger(I) == I
assert qapply(I * O) == O
assert qapply(O * I) == O
for n in [2, 3, 5]:
assert represent(IdentityOperator(n)) == eye(n)
def test_outer_product():
k = Ket('k')
b = Bra('b')
op = OuterProduct(k, b)
assert isinstance(op, OuterProduct)
assert isinstance(op, Operator)
assert op.ket == k
assert op.bra == b
assert op.label == (k, b)
assert op.is_commutative is False
op = k*b
assert isinstance(op, OuterProduct)
assert isinstance(op, Operator)
assert op.ket == k
assert op.bra == b
assert op.label == (k, b)
assert op.is_commutative is False
op = 2*k*b
assert op == Mul(Integer(2), k, b)
op = 2*(k*b)
assert op == Mul(Integer(2), OuterProduct(k, b))
assert Dagger(k*b) == OuterProduct(Dagger(b), Dagger(k))
assert Dagger(k*b).is_commutative is False
#test the _eval_trace
assert Tr(OuterProduct(JzKet(1, 1), JzBra(1, 1))).doit() == 1
# test scaled kets and bras
assert OuterProduct(2 * k, b) == 2 * OuterProduct(k, b)
assert OuterProduct(k, 2 * b) == 2 * OuterProduct(k, b)
# test sums of kets and bras
k1, k2 = Ket('k1'), Ket('k2')
b1, b2 = Bra('b1'), Bra('b2')
assert (OuterProduct(k1 + k2, b1) ==
OuterProduct(k1, b1) + OuterProduct(k2, b1))
assert (OuterProduct(k1, b1 + b2) ==
OuterProduct(k1, b1) + OuterProduct(k1, b2))
assert (OuterProduct(1 * k1 + 2 * k2, 3 * b1 + 4 * b2) ==
3 * OuterProduct(k1, b1) +
4 * OuterProduct(k1, b2) +
6 * OuterProduct(k2, b1) +
8 * OuterProduct(k2, b2))
def test_operator_dagger():
A = Operator('A')
B = Operator('B')
assert Dagger(A*B) == Dagger(B)*Dagger(A)
assert Dagger(A + B) == Dagger(A) + Dagger(B)
assert Dagger(A**2) == Dagger(A)**2
def test_differential_operator():
x = Symbol('x')
f = Function('f')
d = DifferentialOperator(Derivative(f(x), x), f(x))
g = Wavefunction(x**2, x)
assert qapply(d*g) == Wavefunction(2*x, x)
assert d.expr == Derivative(f(x), x)
assert d.function == f(x)
assert d.variables == (x,)
assert diff(d, x) == DifferentialOperator(Derivative(f(x), x, 2), f(x))
d = DifferentialOperator(Derivative(f(x), x, 2), f(x))
g = Wavefunction(x**3, x)
assert qapply(d*g) == Wavefunction(6*x, x)
assert d.expr == Derivative(f(x), x, 2)
assert d.function == f(x)
assert d.variables == (x,)
assert diff(d, x) == DifferentialOperator(Derivative(f(x), x, 3), f(x))
d = DifferentialOperator(1/x*Derivative(f(x), x), f(x))
assert d.expr == 1/x*Derivative(f(x), x)
assert d.function == f(x)
assert d.variables == (x,)
assert diff(d, x) == \
DifferentialOperator(Derivative(1/x*Derivative(f(x), x), x), f(x))
assert qapply(d*g) == Wavefunction(3*x, x)
# 2D cartesian Laplacian
y = Symbol('y')
d = DifferentialOperator(Derivative(f(x, y), x, 2) +
Derivative(f(x, y), y, 2), f(x, y))
w = Wavefunction(x**3*y**2 + y**3*x**2, x, y)
assert d.expr == Derivative(f(x, y), x, 2) + Derivative(f(x, y), y, 2)
assert d.function == f(x, y)
assert d.variables == (x, y)
assert diff(d, x) == \
DifferentialOperator(Derivative(d.expr, x), f(x, y))
assert diff(d, y) == \
DifferentialOperator(Derivative(d.expr, y), f(x, y))
assert qapply(d*w) == Wavefunction(2*x**3 + 6*x*y**2 + 6*x**2*y + 2*y**3,
x, y)
# 2D polar Laplacian (th = theta)
r, th = symbols('r th')
d = DifferentialOperator(1/r*Derivative(r*Derivative(f(r, th), r), r) +
1/(r**2)*Derivative(f(r, th), th, 2), f(r, th))
w = Wavefunction(r**2*sin(th), r, (th, 0, pi))
assert d.expr == \
1/r*Derivative(r*Derivative(f(r, th), r), r) + \
1/(r**2)*Derivative(f(r, th), th, 2)
assert d.function == f(r, th)
assert d.variables == (r, th)
assert diff(d, r) == \
DifferentialOperator(Derivative(d.expr, r), f(r, th))
assert diff(d, th) == \
DifferentialOperator(Derivative(d.expr, th), f(r, th))
assert qapply(d*w) == Wavefunction(3*sin(th), r, (th, 0, pi))
def test_eval_power():
from sympy.core import Pow
from sympy.core.expr import unchanged
O = Operator('O')
U = UnitaryOperator('U')
H = HermitianOperator('H')
assert O**-1 == O.inv() # same as doc test
assert U**-1 == U.inv()
assert H**-1 == H.inv()
x = symbols("x", commutative = True)
assert unchanged(Pow, H, x) # verify Pow(H,x)=="X^n"
assert H**x == Pow(H, x)
assert Pow(H,x) == Pow(H, x, evaluate=False) # Just check
from sympy.physics.quantum.gate import XGate
X = XGate(0) # is hermitian and unitary
assert unchanged(Pow, X, x) # verify Pow(X,x)=="X^x"
assert X**x == Pow(X, x)
assert Pow(X, x, evaluate=False) == Pow(X, x) # Just check
n = symbols("n", integer=True, even=True)
assert X**n == 1
n = symbols("n", integer=True, odd=True)
assert X**n == X
n = symbols("n", integer=True)
assert unchanged(Pow, X, n) # verify Pow(X,n)=="X^n"
assert X**n == Pow(X, n)
assert Pow(X, n, evaluate=False)==Pow(X, n) # Just check
assert X**4 == 1
assert X**7 == X
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